On full-separating sets in graphs

2024-07-15

arXiv.org

AI Summary

This paper addresses the domination-based vertex identification problem in graph theory and introduces, for the first time, the novel separation property of *full separability*, leading to the definitions of *fully separable dominating codes* (FD-codes) and *fully separable total dominating codes* (FTD-codes)—both requiring that any two distinct vertices be uniquely distinguished by their open neighborhoods. Using graph-theoretic analysis, hypergraph covering modeling, and computational complexity theory, we establish necessary and sufficient conditions for existence, derive tight upper and lower bounds on minimum code sizes, and clarify relationships with classical identifying codes. We prove that the minimum sizes of FD- and FTD-codes differ by at most one, and that deciding their equivalence is NP-complete. Moreover, computing minimum FD- and FTD-codes is shown to be NP-hard. Exact minimum sizes are determined for paths, cycles, spider graphs, and other families; several theoretical lower bounds are proven tight, and extremal graph structures are characterized.

Technology Category

• Constraint Satisfaction and Optimization: Distributed CSP/Optimization
• Search and Optimization: Combinatorial Optimization
• Knowledge Representation and Reasoning: Computational Complexity of Reasoning

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Social Networks and Social Media: Influence propagation, information diffusion, and the prediction on networks
• Web Mining and Content Analysis: Models for Web evolution

Abstract

Several different types of identification problems have been already studied in the literature, where the objective is to distinguish any two vertices of a graph by their unique neighborhoods in a suitably chosen dominating or total-dominating set of the graph, often referred to as a emph{code}. To study such problems under a unifying point of view, reformulations of the already studied problems in terms of covering problems in suitably constructed hypergraphs have been provided. Analyzing these hypergraph representations, we introduce a new separation property, called emph{full-separation}, which has not yet been considered in the literature so far. We study it in combination with both domination and total-domination, and call the resulting codes emph{full-separating-dominating codes} (or emph{FD-codes} for short) and emph{full-separating-total-dominating codes} (or emph{FTD-codes} for short), respectively. We address the conditions for the existence of FD- and FTD-codes, bounds for their size and their relation to codes of the other types. We show that the problems of determining an FD- or an FTD-code of minimum cardinality in a graph is NP-hard. We also show that the cardinalities of minimum FD- and FTD-codes differ by at most one, but that it is NP-complete to decide if they are equal for a given graph in general. We find the exact values of minimum cardinalities of the FD- and FTD-codes on some familiar graph classes like paths, cycles, half-graphs and spiders. This helps us compare the two codes with other codes on these graph families thereby exhibiting extremal cases for several lower bounds.

Problem

• Introduces full-separation codes combining domination and separation properties in graphs
• Studies existence conditions and size bounds for FD-codes and FTD-codes
• Proves minimum cardinality determination for these codes is NP-hard

Innovation

• Introducing full-separation property for graph codes
• Studying codes with both domination and separation properties
• Proving NP-hardness of minimum cardinality code problems